Learning periodic motions from human demonstrations using transverse contraction analysis

In this paper, an algorithm called transverse contracting dynamic system primitive (CDSP) to learn the dynamics of periodic motions from demonstrations is presented. Learning motion plans is essential in making robot programming possible by non-expert programmers as well as realizing human-robot collaboration. The complex periodic motion of the human arm is generated by an orbitally stable closed-loop dynamical system. To capture the complexity, a neural network (NN) model is used to represent the dynamics of the human arm motion states. To take into consideration the orbitally stable nature of the human arm motion dynamics, the unknown motion model learning is subjected to transverse contraction analysis constraints. To learn the model parameters, an optimization problem is formulated by relaxing the non-convex transverse contraction constraints using sum of squares (SOS) programming. The NN model is approximated by using a polynomial approximation of the activation function in order to pose the optimization problem as an SOS problem. A negative gradient of a repulsive potential function is added to the learned transverse contracting model to incorporate obstacle avoidance. Experimental results indicate that the model learning via CDSP can generate periodic motions learned from human demonstrated data recorded using Microsoft Kinect sensor. The CDSP algorithm is able to adapt to situations for which the demonstrations are not available, e.g., an obstacle placed in the path. The comparison between the CDSP algorithm and rhythmic dynamic movement primitives (DMPs) shows that the CDSP algorithm performs better in terms of periodic trajectory reproduction accuracy and the time taken by the reproductions to enter the periodic orbit from random initial conditions.